Classical Mechanics Foundations: Quantities, Units, and Motion as a Model

Mechanics is the study of motion and the causes of motion, but in practice it is a modeling activity: we represent real situations using a small set of measurable quantities and simple rules that connect them. The goal is not to describe every detail of reality, but to build a model that is accurate enough for the question we are asking.

1) Core physical quantities (and scalar vs vector)

Throughout classical mechanics, we repeatedly use a small “toolbox” of physical quantities. Each quantity has a meaning, a unit, and (often) a direction.

Common quantities

• Position (x, y, z or vector \vec r): where an object is relative to an origin.
• Time (t): when something happens; used to describe change.
• Mass (m): how much matter/inertia an object has (how strongly it resists changes in motion).
• Force (\vec F): an interaction that can change motion (push/pull), has direction.
• Energy (E): a bookkeeping quantity for the ability to do work or cause change; comes in many forms but shares the same unit.
• Momentum (\vec p): “motion quantity” combining mass and velocity; has direction.

Scalars vs vectors

Scalars have magnitude only. Vectors have magnitude and direction. This distinction matters because vectors must be added using vector rules (direction matters), while scalars add normally.

Quantity	Type	Example
Time t	Scalar	5 s + 3 s = 8 s
Mass m	Scalar	2 kg + 1 kg = 3 kg Continue in our app. Listen to the audio with the screen off. Earn a certificate upon completion. Over 5000 courses for you to explore! Or continue reading below... Download the app
Energy E	Scalar	10 J + 4 J = 14 J
Position \vec r	Vector	“3 m east” is different from “3 m west”
Force \vec F	Vector	Two equal pushes in opposite directions can cancel
Momentum \vec p	Vector	Same speed, opposite direction → opposite momentum

Simple vector example: If you walk 2 m east, then 2 m west, your total displacement (a vector) is 0 m, even though your total distance traveled (a scalar) is 4 m.

2) SI units and dimensional analysis

To make models testable, every quantity must be expressed in units. In mechanics we mainly use SI base units and a few derived units.

Key SI units used in mechanics

• Length: meter (m)
• Time: second (s)
• Mass: kilogram (kg)

From these, many important derived units follow:

• Velocity: m/s
• Acceleration: m/s^2
• Force (newton): N = kg·m/s^2
• Energy (joule): J = N·m = kg·m^2/s^2
• Momentum: kg·m/s

Dimensional analysis: checking equations by units

Dimensional analysis means verifying that both sides of an equation have the same dimensions (built from kg, m, s). It cannot prove an equation is correct, but it can quickly show when an equation is impossible.

Step-by-step unit check example: Suppose someone claims the kinetic energy is E = m v.

1. Write units of each symbol: [m] = kg, [v] = m/s.
2. Compute units of the right-hand side: [m v] = kg·m/s.
3. Compare with energy units: [E] = J = kg·m^2/s^2.
4. They do not match, so E = m v cannot be correct.

Unit check that works: For momentum \vec p = m \vec v, the units are kg·m/s, which matches momentum.

Activity: quick unit conversions (practice)

Use the idea “multiply by 1” with conversion factors.

• Convert 72 km/h to m/s
  1. Start: 72 km/h
  2. Use 1 km = 1000 m and 1 h = 3600 s
  3. 72 km/h × (1000 m / 1 km) × (1 h / 3600 s) = 72 × 1000 / 3600 m/s = 20 m/s
• Convert 250 cm to m
  1. 250 cm × (1 m / 100 cm) = 2.5 m
• Convert 0.30 g to kg
  1. 0.30 g × (1 kg / 1000 g) = 3.0×10^-4 kg

Activity: spot the unit mistake

Each statement below contains a unit error. Identify it.

• F = 12 kg (force written in kilograms)

• v = 15 m/s^2 (velocity written in acceleration units)

• E = 40 N (energy written in newtons instead of joules)

Fixing them means choosing the correct unit: force in N, velocity in m/s, energy in J.

3) Coordinate systems, sign conventions, and reference frames

To turn motion into numbers, you must choose a coordinate system and a reference frame. This choice is part of the model. A good choice makes the math simpler and reduces mistakes.

Coordinates and axes

In one dimension, we often describe position with a single coordinate x on a line. You must choose:

• Origin: where x = 0 is located.
• Positive direction: which way counts as +x.

In two or three dimensions, we use axes (x, y, z) and represent position as a vector \vec r.

Sign conventions

Signs are not “built into nature”; they come from your axis choice. Once chosen, you must apply them consistently.

• If +x is to the right, then motion to the left has negative velocity.
• If you choose +y upward, then gravity near Earth points in the negative y direction.

Practical tip: Before calculating, write a tiny sketch with an arrow showing the positive axis direction.

Reference frames: what changes and what does not

A reference frame is the viewpoint from which you measure positions and times (often tied to an observer or a lab). Changing frames can change measured values of some quantities.

• Depends on frame: position, displacement, velocity, momentum (because they depend on motion relative to the frame).
• Typically frame-independent in basic mechanics: mass (for everyday speeds), elapsed time for the same clock in the same situation, and the form of physical laws when using appropriate frames.

Example: A passenger on a train measures a ball’s velocity relative to the train. A person standing on the ground measures a different velocity for the same ball because the train itself is moving relative to the ground.

Activity: choose axes for a scenario

Scenario: A box slides down a ramp that slopes downward to the right.

1. Draw the ramp as a line slanting down to the right.
2. Choose an axis +x along the ramp pointing downhill (to the right). This often makes the motion one-dimensional.
3. Optionally choose +y perpendicular to the ramp (useful later when separating forces).
4. State your choice in words: “+x is down the ramp; origin at the top.”

This choice does not change the physics, but it can simplify the equations you write.

4) Model vs reality: assumptions and approximations

Real objects are complex: they have shape, spin, deform, heat up, and interact with air. Mechanics often begins with simplified models that capture the dominant effects.

Point particle approximation

In many problems, an object’s size and shape do not matter for the question being asked. Then we treat it as a point particle: all its mass is considered concentrated at a single point, and we track only its position over time.

• Good when: the object is small compared to distances traveled (e.g., a car driving 1 km; its 4 m length is not important for basic motion).
• Not good when: rotation, tipping, or size matters (e.g., a rolling cylinder, a gymnast rotating, a ladder slipping).

Ignoring air resistance (when appropriate)

Air resistance can be small or large depending on speed, shape, and area.

• Often reasonable to ignore: a dense ball thrown at moderate speed over a short distance (first estimate).
• Often not reasonable to ignore: a feather, a sheet of paper, or high-speed motion where drag grows strongly.

Ignoring air resistance is not “lying”; it is stating a modeling assumption. If predictions disagree with measurements, you may need a better model that includes drag.

What assumptions mean in practice

Every model has a “validity range.” When you write assumptions, you are defining the conditions under which your results should be trusted.

• Assumption: “Treat the object as a point particle.” Meaning: we ignore rotation and size effects.
• Assumption: “Neglect air resistance.” Meaning: only gravity (and maybe contact forces) are included.
• Assumption: “Use a flat Earth approximation.” Meaning: over small distances, curvature is negligible.

Mini-checklist before solving a mechanics problem

• What quantities are relevant (position, time, mass, force, energy, momentum)?
• Which are vectors? What directions matter?
• What units will you use? Are they SI?
• What coordinate system and sign convention will you adopt?
• What reference frame are measurements relative to?
• What assumptions are you making (point particle, no air resistance, etc.)?

Activity: find and fix unit inconsistencies in equations

For each proposed equation, decide whether the units can match. If not, explain why.

• x = v + t (adding m/s and s cannot produce m)

• F = m a (units: kg·m/s^2 = N, consistent)

• E = F / x (units: N/m not equal to J)